A099444 -id:A099444 - cp's OEIS frontend

A099445 An Alexander sequence for the Miller Institute knot.

1, 3, 6, 12, 25, 54, 117, 252, 542, 1167, 2514, 5415, 11662, 25116, 54093, 116502, 250913, 540396, 1163862, 2506635, 5398594, 11627067, 25041462, 53932332, 116155217, 250165974, 538787805, 1160398812, 2499175726, 5382528183

Offset: 0

Paul Barry, Oct 16 2004

The denominator is a parameterization of the Alexander polynomial for the knot 6_2 (Miller Institute knot). The g.f. is the image of that of Fib(2n+2) under the modified Chebyshev transform A(x)->(1/(1+x^2)^2)A(x/(1+x^2)).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dror Bar-Natan, The Rolfsen Knot Table
Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-1).

Cf. A001906.

I:=[1,3,6,12,25,54,117,252]; [n le 8 select I[n] else 3*Self(n-1)-3*Self(n-2)+3*Self(n-3)-Self(n-4) : n in [1..50]]; // Vincenzo Librandi, Feb 12 2014

CoefficientList[Series[(1 - x) (x + 1) (x^2 + 1)/(x^4 - 3 x^3 + 3 x^2 - 3 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
LinearRecurrence[{3,-3,3,-1},{1,3,6,12,25},30] (* Harvey P. Dale, Jun 24 2018 *)

Vec(-(x-1)*(x+1)*(x^2+1)/(x^4-3*x^3+3*x^2-3*x+1) + O(x^100)) \\ Colin Barker, Feb 10 2014

G.f.: -(x-1)*(x+1)*(x^2+1) / (x^4-3*x^3+3*x^2-3*x+1). - Colin Barker, Feb 10 2014
a(n) = A099444(n)-A099444(n-2).
a(n) = 3*a(n-1)-3*a(n-2)+3*a(n-3)-a(n-4) for n>4. - Colin Barker, Feb 10 2014

G.f. corrected by Colin Barker, Feb 10 2014

cp's OEIS Frontend

A099445 An Alexander sequence for the Miller Institute knot.

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